{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Billiards\n",
    "\n",
    "[![Google Collab Book](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/tum-pbs/PhiFlow/blob/develop/examples/particles/Billiards.ipynb)\n",
    "\n",
    "This example simulates Billiard balls and collisions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%pip install --quiet phiflow\n",
    "from phi.torch.flow import *\n",
    "# from phi.flow import *  # If JAX is not installed. You can use phi.torch or phi.tf as well."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's create a cue ball and the typical billiards triangle!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 864x360 with 1 Axes>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 864x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def create_balls(cue_pos=vec(x=.1, y=.5), billiard_layers=4, radius=.03):\n",
    "    coords = [cue_pos]\n",
    "    for i in range(billiard_layers):\n",
    "        for j in range(i + 1):\n",
    "            coords.append(vec(x=i * 2 * radius + 0.5, y=j * 2 * radius + 0.5 - i * radius * 0.7))\n",
    "    return Sphere(stack(coords, instance('ball')), radius=radius)\n",
    "\n",
    "balls = create_balls()\n",
    "plot(balls, color=math.range(instance(balls)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we define the dynamics consisting of linear movement and collisions.\n",
    "We store the velocities in the `values` of the field.\n",
    "The impact dynamics depend on the relative distances and velocities of the balls, which we get using `math.pairwise_differences`.\n",
    "Summing the neighbor contributions then amounts to summing over the dual dimension."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "@jit_compile\n",
    "def physics_step(v: PointCloud, dt: float, elasticity=0.8):\n",
    "    v_next = advect.points(v, v, dt)\n",
    "    x_diff = math.pairwise_differences(v_next.points)\n",
    "    dist = math.vec_length(x_diff, eps=1e-4)  # eps to avoid NaN during backprop of sqrt\n",
    "    rel_v = -math.pairwise_differences(v.values)\n",
    "    dist_dir = -math.safe_div(x_diff, dist)\n",
    "    projected_v = dist_dir.vector * rel_v.vector\n",
    "    has_impact = (projected_v < 0) & (dist < 2 * v.geometry.radius)\n",
    "    impulse = -(1 + elasticity) * .5 * projected_v * dist_dir\n",
    "    radius_sum = v.geometry.radius + rename_dims(v.geometry.radius, instance, dual)\n",
    "    impact_time = math.safe_div(dist - radius_sum, projected_v)\n",
    "    x_inc_contrib = math.sum(math.where(has_impact, math.minimum(impact_time - dt, 0) * impulse, 0), dual)\n",
    "    v = v.with_elements(v.geometry.shifted(x_inc_contrib))\n",
    "    v += math.sum(math.where(has_impact, impulse, 0), dual)\n",
    "    return advect.points(v, v, dt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's give the cue ball a starting velocity and run the simulation!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "v0 = math.scatter(math.zeros(balls.shape), indices=vec(ball=0), values=vec(x=3, y=0))\n",
    "initial_state = Field(balls, v0, 0)\n",
    "trj = iterate(physics_step, batch(t=60), initial_state, dt=0.003, substeps=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<matplotlib.animation.FuncAnimation at 0x1f3a44ee1f0>"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(trj.geometry, animate='t', color=math.range(instance(balls)))"
   ]
  }
 ],
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   "display_name": "phiflow2",
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